International Journal of Scientific & Engineering Research, Volume 8, Issue 3, March-2017 ISSN 2229-5518

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Asymmetric Fission from Egg Shape Deformation and Comparison of Coulomb and Surface Ener-gies for Fission Fragment Mass Combinations

Mahendra Kushwaha1, 2

1Department of Physics, University of Mumbai, Vidyanagri, Santacruz (E), Mumbai – 400 098, India

2Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai – 400 085, India

Abstract - Asymmetric fission from an egg shape deformation is studied for 235

U nucleus. For neck formation in the fissioning process, the

place of least curvature of deformed nucleus is taken into account. Geometry of an egg shape is expressed in terms of its major and minor

axes, and variation of Coulomb and surface energies studied. In the symmetric and asymmetric fission fragment mass combinations,

significant variation of about 4 to 10 MeV in Coulomb energy is observed which can affect the fission barrier height. For the asymmetrically

deformed nucleus like an egg, the shell effects are discussed. It is found that parity is not a good quantum number for asymmetric shape

hence the asymmetric shape like an egg is supposed to evolve at the crossing of opposite parity levels.

Keywords — Fission, Asymmetric, Egg shape, geometry, Barrier, Coulomb, Surface, Shell effects

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1 INTRODUCTION

UCLEAR fission was discovered around three quarters of a century

ago in which the splitting of a heavy nucleus into various fragments

was observed [1-3]. The general features of fission reactions were well

understood both experimentally and theoretically but still it continued to

throw up new challenges. One of the most persistent puzzles in the fission

process is the asymmetric mass distributions of fission fragments. In-

volvement of very large number of nucleons and both macroscopic and

microscopic features are expected to influence the fission phenomena [3].

Macroscopic features are basically of classical nature which largely de-

termines the energetics of the fission phenomena. The liquid drop model

[3-6] is a very well established classical model which describes the defor-

mation process in terms of nuclear surface and Coulomb energy and cor-

responding fission barrier due to competition between surface and Cou-

lomb energies [5-6]. On the other hand, asymmetric mass distribution of

fission fragments pointed towards the more microscopic quantum effects

[7-10]. These quantum effects describe the distribution of nucleons in the

nuclear shells for a deformed potential.

2 THEORETICAL DESCRIPTION

Shape changes involved in the transition from a single spherical

nuclear system to two separated daughter fragments is considered to be

as a slow process. In the fissioning process however starting from a

spherical shape, at some point of deformation the nucleus has to assume

an asymmetrical shape in the fissioning direction. Some common defor-

mation shapes [3, 11-13] are like ellipsoid, dumbbell, pear shape etc. One

of the simplest asymmetrical shapes is like an egg which is normally not

studied. This egg shape is obtained basically from two hemi - ellipsoids of

revolution about the major axes – a1 and a2 with equal minor axis - b, by

joining the equal circular faces as seen in fig.1. With increasing elongation

in the fissioning direction the major axes - a1 and a2 increases and corres-

ponding minor axis - b decrease so as to conserve the total volume.

Fig.1: Schematic of an asymmetric egg shape (with minor axis b, and

major axes a1 and a2) and deformation process via an egg shape. 1 and 2

are the two point charges within egg volume.

Fission occurs when the joint is snapped as a result of the total surface

energy increasing beyond the total surface energy of the spherical shape.

In the fissioning process it is important that the cleavage starts in this pro-

ceedings of elongation at a point where the surface curvature is least and

N

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International Journal of Scientific & Engineering Research, Volume 8, Issue 3, March-2017 ISSN 2229-5518

IJSER © 2017

http://www.ijser.org

it is at the joint of the two hemi-ellipsoids.

3 NUMERICAL CALCULATIONS AND RESULTS

In the present paper, the numerical calculations are performed

for evaluation of the Coulomb and surface energies for the nearly symme-

tric and highly asymmetric mass combinations of fission fragments of 235

U

nucleus. The Coulomb energy is evaluated by performing six-dimensional

integration as:

Where

The surface energy however has an analytic form as:

Where

And surface area is defined as

Fig. 2: Surface energy of ellipsoid and an egg shaped nucleus.

It is found that, final asymmetrical shape of an egg does not start at the

beginning of the deformation process rather it starts when the volume

conservation, cleavage point (b < a1 < a2) and a1 / a2 conditions are satis-

fied simultaneously. However this occurs only after slight deformation has

taken place. Variation of surface and Coulomb energy as a function of

minor axis - b, for symmetric to asymmetric fission into various fragments

mass combinations (as 80 + 155, 85 + 150 to 115 + 120) is shown in fig. 2

and fig. 3 respectively. The value of parameter - b decreasing from the

sphere on these plots corresponds to the point where asymmetry starts

developing and the last smallest value is the value where the surface

energy equals that of the fragments (the saddle point). Comparison of

surface energy between symmetric and asymmetric fission fragment mass

combinations is shown in fig. 2. As a function of minor axes - b, the sur-

face energy of symmetrical and asymmetrical mass combinations is seen

to overlap and behaves similarly. Observed linear relation between the

ratio of major axes a1 / a2 and minor axis - b is represented in fig. 4.

Fig. 3: Coulomb energy of fission fragments mass combinations as a

function of minor axis - b.

Fig. 4: Variation of ratio of major axes a1/a2 with minor axis – b.

Most importantly from figure - 3, it is found that, the Coulomb energy for

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International Journal of Scientific & Engineering Research, Volume 8, Issue 3, March-2017 ISSN 2229-5518

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asymmetric fragment mass combinations is seen to decreases from that of

the symmetric fission fragments. Around 4 to 10 MeV energy differences

are observed in between asymmetric and symmetric fission fragment

mass combinations and this behavior can affect the fission barrier height.

This also indicates that large amount of energy shell corrections are re-

quired for asymmetric deformations.

On the other hand, geometry of the egg shape in terms of com-

bination of two hemi- ellipsoid is expressed in terms of major and minor

axes and deformation parameter - ε [14] as;

(Along major axis - a1)

and

(Along major axis - a2)

Where R = 7.53 fm (Radius of spherical nucleus 235

U)

Fig. 5 (A & B): Variation of minor axis – b with deformation parameter ɛ12

and ɛ21.

Here ɛ12 and ɛ21 have the same values. The relation of these deformation

parameters with the minor axis - b is shown in fig. 5A and fig. 5B respec-

tively.

The mass asymmetry in fission is greatly influenced by the dis-

tribution of nucleons in the nuclear shells. It is reported that the liquid drop

energy surface is stable with respect to the asymmetric distortions [3, 15-

16]. For large P2 distortions, the liquid drop energy surface becomes quite

soft with respect to distortions that involve a certain combinations of P3

and P5 deformations [15-16]. It is thus possible that at certain deforma-

tions, shell effects might overcome the liquid drop stability with respect to

asymmetric distortions [16].

To study the cause of asymmetric distortions, the Nilsson dia-

gram [17] of deformed nucleus is employ. In the Nilsson diagram it is

found that the asymmetry can be start within 0.0 to 0.3 of the range of

deformation - ε and observed that there are some points where the energy

shells those having same spin (like 1/2, 3/2, 5/2, 7/2, 9/2 etc.) but opposite

parity (solid and dotted lines) cross each other. These are the points (the

point of crossing of solid and dotted lines which have same spin but oppo-

site parity) where mixed parity exists and asymmetries being developed in

the form of asymmetric shape like an egg. These points are studied in the

Nilsson diagram of deformed nucleus for Z > 82 and also for N ≥ 126 re-

spectively.

4 SUMMARY AND CONCLUSION

The deformations and fission fragment mass asymmetry is discussed

by combining both macroscopic and microscopic methods. Asymmetric

fission via an egg shape deformation is studied. One important result here

is that the cleavage starts at that point where the surface curvature is least

and this is the place of joint of two hemi-ellipsoids. The present perception

says that the lesser eccentricity ellipsoid, a1/b part of the egg become the

smaller fragment while the bigger eccentricity part; a2/b become the larger

fragment. The prevailing perception in the available literature does not

describe the phenomena of asymmetric fission via the formation of an egg

shape and they tend to describes the asymmetric fission through some

neck formation with one big blob on one side while a smaller one on the

other side. So far this aspect of P3 distortions for asymmetric fission has

not been highlighted for this egg shaped deformation and present work

indicates that higher amount of shell corrections to be applicable in the

asymmetric fission. In the Nilsson diagram it is found that the asymmetry

starts at the points where the energy shells of same spin but opposite

parity (solid and dotted lines) cross each other. These are the points

where mixed parity exists and asymmetry being developed in the form of

asymmetric shape like an egg.

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International Journal of Scientific & Engineering Research, Volume 8, Issue 3, March-2017 ISSN 2229-5518

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ACKNOWLEDGEMENT

Author thanks Dr. Arun K. Jain, NPD-BARC, Mumbai for needful

discussion for calculations and also thank to Department of Science and

Technology (DST), New Delhi - India, for financial support. Thanks are due

to Bhabha Atomic Research Centre for providing necessary facility for the

work.

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